Optimal. Leaf size=48 \[ \frac{e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac{b e \tanh ^{-1}(c+d x)}{2 d}+\frac{b e x}{2} \]
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Rubi [A] time = 0.0344056, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6107, 12, 5916, 321, 206} \[ \frac{e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac{b e \tanh ^{-1}(c+d x)}{2 d}+\frac{b e x}{2} \]
Antiderivative was successfully verified.
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Rule 6107
Rule 12
Rule 5916
Rule 321
Rule 206
Rubi steps
\begin{align*} \int (c e+d e x) \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int e x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac{(b e) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac{b e x}{2}+\frac{e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac{(b e) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac{b e x}{2}-\frac{b e \tanh ^{-1}(c+d x)}{2 d}+\frac{e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0464528, size = 77, normalized size = 1.6 \[ \frac{e \left (2 a c^2+4 a c d x+2 a d^2 x^2+b \log (-c-d x+1)-b \log (c+d x+1)+2 b (c+d x)^2 \tanh ^{-1}(c+d x)+2 b c+2 b d x\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 107, normalized size = 2.2 \begin{align*}{\frac{a{x}^{2}de}{2}}+xace+{\frac{a{c}^{2}e}{2\,d}}+{\frac{d{\it Artanh} \left ( dx+c \right ){x}^{2}be}{2}}+{\it Artanh} \left ( dx+c \right ) xbce+{\frac{{\it Artanh} \left ( dx+c \right ) b{c}^{2}e}{2\,d}}+{\frac{bex}{2}}+{\frac{bec}{2\,d}}+{\frac{be\ln \left ( dx+c-1 \right ) }{4\,d}}-{\frac{be\ln \left ( dx+c+1 \right ) }{4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.966486, size = 153, normalized size = 3.19 \begin{align*} \frac{1}{2} \, a d e x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{artanh}\left (d x + c\right ) + d{\left (\frac{2 \, x}{d^{2}} - \frac{{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac{{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b d e + a c e x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b c e}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31053, size = 169, normalized size = 3.52 \begin{align*} \frac{2 \, a d^{2} e x^{2} + 2 \,{\left (2 \, a c + b\right )} d e x +{\left (b d^{2} e x^{2} + 2 \, b c d e x +{\left (b c^{2} - b\right )} e\right )} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.29738, size = 95, normalized size = 1.98 \begin{align*} \begin{cases} a c e x + \frac{a d e x^{2}}{2} + \frac{b c^{2} e \operatorname{atanh}{\left (c + d x \right )}}{2 d} + b c e x \operatorname{atanh}{\left (c + d x \right )} + \frac{b d e x^{2} \operatorname{atanh}{\left (c + d x \right )}}{2} + \frac{b e x}{2} - \frac{b e \operatorname{atanh}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\c e x \left (a + b \operatorname{atanh}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2337, size = 192, normalized size = 4. \begin{align*} \frac{b d^{2} x^{2} e \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 2 \, a d^{2} x^{2} e + 2 \, b c d x e \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 4 \, a c d x e + b c^{2} e \log \left (d x + c + 1\right ) - b c^{2} e \log \left (-d x - c + 1\right ) + 2 \, b d x e - b e \log \left (d x + c + 1\right ) + b e \log \left (-d x - c + 1\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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